Classical weight one forms in a Hida family of Hilbert cusp forms with parallel weights

Let F be a totally real field and p an odd prime. We consider a primitive p-ordinary Hida family of Hilbert cusp forms with parallel weights defined over F. It is proved by Hida that the specialization of such a family at any arithmetic point of weight greater than or equal to two is classical, namely, a holomorphic Hilbert cusp form. However, this is not always the case for weight one specializations. 

In the case of elliptic cusp forms, Ghate and Vatsal show that such a family admits infinitely many classical weight one specializations if and only if it is a CM family. Further, the number of classical weight one specializations inside a non-CM family is bounded by an explicit constant due to Dimitrov and Ghate. 

In this talk, I will discuss a generalization of these works to the case of Hilbert modular forms with parallel weights.